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Retaining perturbation thermal diffusion givesThe displaced interfacial temperature and Stefan condition give the displayed implicit dispersion relation. In the capillarity-free limit its growing branch is exactly $\sigma =V\alpha$, which checks the limits of the small-wavelength thermal approximation for a solidification front.

For a paired non-self-conjugate momentum, rotate the Ising-chain Nambu spinor by an angle with $\cos \theta =(g-\cos k)/r_k$, $\sin \theta =-\sin k/r_k$ and $r_k=\sqrt{1+g^2-2g\cos k}$. Opposite angles at $k,-k$ preserve the canonical anticommutation relations. The result is $H={\sum }_k{ϵ}_k({\gamma }_k^{†}{\gamma }_k-1/2)$. Self-paired momenta have no pairing and are handled directly as number levels. Periodic spin-chain sectors additionally constrain fermion parity and allowed mode quantization.

For positive ferromagnetic exchange and nonnegative transverse field, the Ising-chain quasiparticle dispersion closes its gap at $g=1$, with ${ϵ}_k\sim 2J|k|$. This is the quantum phase transition between the model's ordered and field-polarized regimes. The exact dispersion identifies the gapless scale; computing its order parameter additionally requires ground-state correlations.

With translation of a distribution defined by $⟨{\tau }_{h}u,\phi ⟩=⟨u,\phi (\cdot +h)⟩$, differentiation of the test function gives $\frac{d}{dh}{\tau }_{h}u=-{\tau }_h{u}^{\prime }$ in weak convergence of distributions. The Taylor theorem shows convergence in the space of test functions, including a common compact support. Thus $({\tau }_{-h}u-u)/h\to u^{\prime }$; fixing the translation convention prevents a sign error.

Adjoin a new point $\omega$ fixed by a transitive group $G$. A one-point extension is a transitive group on the enlarged set whose stabilizer of $\omega$ is exactly $G$, acting on the remaining points in the prescribed way. Its order is $(n+1)|G|$ in the finite degree-$n$ case. The double-coset criterion for a one-point extension gives a concrete generator test.

An action is sharply $t$-transitive if each ordered $t$-tuple of distinct points is carried to any other by exactly one group element. For finite degree $n\ge t$, its order is $n(n-1)\cdots (n-t+1)$ and each ordered-tuple stabilizer is trivial. Sharp one-transitivity is a regular action. Finite sharp two-transitivity gives a regular kernel of a finite sharply two-transitive group.

A regular permutation subgroup acts transitively with trivial point stabilizers. Each ordered pair of points is related by exactly one subgroup element. A finite regular subgroup on $n$ points has order $n$. Normality and characteristicity are additional properties; they do not follow from regularity alone.

An imputation is an efficient, individually rational allocation in a transferable utility game. Its total is the grand-coalition value and each player receives at least its singleton value. The set is nonempty exactly when $v(N)\ge {\sum }_{i}v(\{i\})$, and is then compact. This individual-rationality condition distinguishes the nucleolus from the prenucleolus.

When the imputation set is nonempty, the nucleolus is the unique imputation lexicographically minimizing the vector of excesses of a coalition sorted decreasingly. It minimizes the largest complaint first, then subsequent complaints among ties. It can be computed through successive linear programs that minimize the current maximum excess and retain constraints fixed by prior stages. Its uniqueness is the standard nucleolus theorem. If the core of a cooperative game is nonempty, the nucleolus belongs to it; an empty core does not preclude a nucleolus.

Duality assigns a coalition the grand-coalition value less the value of its complement. For a normalized transferable utility game, the dual is normalized and taking the dual twice recovers the original game. A player's dual marginal contribution at predecessor set $S$ equals its original contribution at the complementary predecessor set in $N\setminus \{i\}$. This preserves the Shapley value.

The transcendence degree is the cardinality of a transcendence basis. It counts independent transcendental parameters. In a finitely generated separably generated field extension, it equals the dimension of the Kähler differentials as a vector space over the extension field.

A transcendence basis $T$ for $L/k$ is separating if $L/k(T)$ is a separable field extension. In this case the $dt$, $t\in T$, form a basis of the Kähler differentials. In characteristic zero every transcendence basis is separating. In positive characteristic the condition on the chosen basis is essential.

If continuously differentiable $F_j,F$ converge uniformly on a compact interval and have uniformly bounded total variation of a function, then $\int b{F}_j^{\prime }\to \int b{F}^{\prime }$ for every continuous $b$. For continuously differentiable test functions this is integration by parts. Uniform approximation by polynomials extends the conclusion to continuous tests using the variation bound.

The product of three circles is a compact smooth three-manifold. Quotient coordinates allow constant commuting vector fields. A rank-two plane field can have dense immersed leaves, as in a dense immersed cylinder in a three-dimensional torus.

For an abelian group $A$, its $p$-primary torsion subgroup is $A\{p\}=\{a:p^{r}a=0\text{ for some }r\ge 0\}$. It is zero exactly when $A[p]=0$, since a nonzero element of $p$-power order has a multiple of order $p$. Multiplication by an integer prime to $p$ is an automorphism on it.

Apply clearing denominators relative to an independent module subset to find $0\ne a$ with $aM$ lying in a finite free module. Multiplication by $a$ is injective on a torsion-free module, giving the required embedding. A submodule need not itself be free over a general integral domain.

Linear axisymmetric compressive modes of a barotropic closure of a razor-thin disk combine radial epicyclic restoration, softened self-gravity and pressure. The off-plane razor-thin Poisson kernel supplies the exponential gravity reduction. Here $k$ is the nonnegative radial wavenumber magnitude. The full linear system also has an axisymmetric geostrophic mode.

At the marginal radial wavenumber, the forced axisymmetric density mode with vortensity has zero squared frequency. Its mass density amplitude is $C_0+C_{1}t-\Omega {\Sigma }_0^2{q}^{\prime }{t}^2$. Thus there is no exponential instability but nonzero conserved vortensity can force quadratic growth, and a homogeneous generalized mode can grow linearly. The spectral boundary $Q=1$ of the Toomre stability criterion must not be mistaken for boundedness of every initial-value solution.

In units where a circular planet has orbital radius and speed one, the particle's relative speed $u$ outside a close encounter satisfies $u^2=3-T$, to leading order in the planet-to-star mass ratio. This follows from the vis-viva equation and $u^2=v^2+1-2h_z$ at the encounter radius. A short conservative encounter changes the direction of this relative velocity while preserving its magnitude.

For bound planet-crossing Kepler orbits, $T=2/(q+Q)+2\sqrt{2qQ/(q+Q)}\cos I$, with the planet radius taken as one. At fixed $q<1$, $T$ is maximized by $I=0$ and $Q=1$. For $2<T<3$ this gives $q\ge (1-\sqrt{3-T}{)}^2/[1+2\sqrt{3-T}-(3-T)]$. This form remains regular at $T=\sqrt{8}$. It is an invariant-based accessibility bound, not a guarantee that a scattering history attains it; for $0<T\le 2$ it gives no positive universal pericentre floor.